Theorem arwhoma 18058

Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwrcl.a	⊢ 𝐴 = (Arrow‘𝐶)
arwhoma.h	⊢ 𝐻 = (Homa‘𝐶)

Assertion

Ref	Expression
arwhoma	⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))

Proof of Theorem arwhoma

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		arwrcl.a	. . . . . . 7 ⊢ 𝐴 = (Arrow‘𝐶)
2		arwhoma.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
3	1, 2	arwval 18056	. . . . . 6 ⊢ 𝐴 = ∪ ran 𝐻
4	3	eleq2i 2826	. . . . 5 ⊢ (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ ∪ ran 𝐻)
5	4	biimpi 216	. . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ∪ ran 𝐻)
6		eqid 2735	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
7	1	arwrcl 18057	. . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)
8	2, 6, 7	homaf 18043	. . . . 5 ⊢ (𝐹 ∈ 𝐴 → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V))
9		ffn 6706	. . . . 5 ⊢ (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)))
10		fnunirn 7246	. . . . 5 ⊢ (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)))
11	8, 9, 10	3syl 18	. . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)))
12	5, 11	mpbid 232	. . 3 ⊢ (𝐹 ∈ 𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧))
13		fveq2 6876	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
14		df-ov 7408	. . . . . 6 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
15	13, 14	eqtr4di 2788	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
16	15	eleq2d 2820	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹 ∈ (𝐻‘𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦)))
17	16	rexxp 5822	. . 3 ⊢ (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
18	12, 17	sylib 218	. 2 ⊢ (𝐹 ∈ 𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦))
19		id 22	. . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦))
20	2	homadm 18053	. . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (doma‘𝐹) = 𝑥)
21	2	homacd 18054	. . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (coda‘𝐹) = 𝑦)
22	20, 21	oveq12d 7423	. . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → ((doma‘𝐹)𝐻(coda‘𝐹)) = (𝑥𝐻𝑦))
23	19, 22	eleqtrrd 2837	. . . 4 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))
24	23	rexlimivw 3137	. . 3 ⊢ (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))
25	24	rexlimivw 3137	. 2 ⊢ (∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))
26	18, 25	syl 17	1 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  Vcvv 3459  𝒫 cpw 4575  ⟨cop 4607  ∪ cuni 4883   × cxp 5652  ran crn 5655   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  dom_acdoma 18033  cod_accoda 18034  Arrowcarw 18035  Hom_achoma 18036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-1st 7988  df-2nd 7989  df-doma 18037  df-coda 18038  df-homa 18039  df-arw 18040

This theorem is referenced by:  arwdm  18060  arwcd  18061  arwhom  18064  arwdmcd  18065  coapm  18084